FWIW, it's been a trend for a bit now for teens who've been "given" just the dad's name to decide to switch to using both last names even if the parents are still together.
If they're under 16 or not ready to make a legal change they just use the double-barreled version as their "known as" name** (which also avoids the paperwork question until she's old enough to assess whether the legal change is worth the hassle of the extra piece of documentation for life). Whether or not others use or respect it may depend on the community, I guess - but I don't see why they wouldn't as surname changes are pretty routine. While there can be moments of forgetting or confusion, most people get used to it just as they do when a name is changed due to marriage, divorce, adoption, etc.
IME it's pretty common for younger children to want the mother's name post-divorce especially if (1) the dad has less than 50% custody or (2) the child is especially close with extended family on the mother's side, and then eventually just because if feels like the fair and egalitarian thing to do. Maybe they do get the idea from adults originally, but that doesn't mean it doesn't become important to them.
What happens if you remarry and take his name, she’ll literally be the only Evans-Johnson in her family. She'd be the only one anyway, because the senior Evans and Johnson are each using their original last names and are unlikely to have another child together.
Why all the angst about what happens when two people with double-barreled names marry? 500 million Hispanophones aren't running around like chickens panicking about it. Coverture (the principle that females do not have an independent legal identity but must be "covered" by a designated male) was historically adopted earlier and more enthusiastically in England than anywhere else in Europe.
** (This is in Scotland, where the child's birth records are indexed under both names and the de facto name has more legal force than in England/Wales, but I'm fairly sure you can do similar elsewhere in the UK as well).